1. Field of the Invention
The present invention relates generally to a system and method for management of a portfolio of financial assets and, more particularly, to large-scale portfolio optimization. Specifically, various embodiments in accordance with the present invention provide a system and method for expected utility maximization in large-scale portfolio optimization.
2. Description of the Prior Art
Portfolio problems are routinely formulated and solved as mean-variance portfolio optimization problems, based on H. Markowitz, Portfolio selection, Journal of Finance, 7(1):77-91, 1952, where expected return and risk of a portfolio are traded off and where portfolio risk is represented as portfolio variance. For example. let R be the random n-vector of asset returns. The mean-variance portfolio optimization problem may then be stated as
            max      ⁢                          ⁢              μ        T            ⁢      x        -                  γ        2            ⁢              x        T            ⁢      Mx                  Ax      =      b        ,          l      ≤      x      ≤      h      where μ=ER is the n-vector of expected asset returns, M=[Mij] is the n×n covariance matrix of asset returns (Mij=cov(Ri, Rj)), γ is the risk aversion parameter, Ax=b are linear constraints and l, and h are lower and upper bounds on asset holdings.
Mean-variance optimization is particularly appropriate when asset returns are distributed according to a multivariate normal distribution, i.e., R≈N(μ, M), because in this case the distribution is fully determined by μ and M only, as all higher moments are zero.
A related concept is expected utility maximization. For example. let u be a monotone increasing and concave utility function of wealth. The expected utility maximization problem may be expressed asmaxE u (1+RTx)Ax=b,l≦x≦h where, given an initial wealth normalized to 1, an end-of-period wealth is given by the random variable W=1+RTx. The particular functional form of the utility function u in the above expression represents an investor preference. Utility functions commonly used are from the hyperbolic absolute risk aversion (HARA) class of utility functions, but also utility functions based on lower partial moments that explicitly penalize outcomes below a certain target wealth are appropriate choices. Often in finance the power function is used, i.e.,
            u      ⁡              (        W        )              =                            W                      1            -            γ                          -        1                    1        -        γ              ,where here γ represents the constant (with respect to wealth) relative risk aversion parameter describing the investor preference towards risk.
Expected utility maximization is a broader concept than mean-variance optimization. Expected utility maximization facilitates the appropriate representation of all higher moments (e.g., skewness, kurtosis, etc.) of the asset return distribution in the portfolio optimization framework. If
            u      ⁡              (        W        )              =          EW      -                        γ          2                ⁢                  var          ⁡                      (            W            )                                ,the mean-variance and the expected utility maximization models are identical. However, any other utility function will yield different results, when asset returns are not multivariate normally distributed. Otherwise, if asset return distributions are multivariate normally distributed, any monotone increasing and concave utility function will yield a mean-variance efficient portfolio.
One expects different results for mean-variance and expected utility maximization for asset returns that are not multivariate normally distributed. The differences in the portfolios may be small as Y. Kroll, H. Levy, and H. Markowitz, Mean-variance versus direct utility maximization, Journal of Finance, 39(1):47-61, 1984, argued.
In practical implementations of mean-variance portfolio optimization, the mean vector pt and the covariance matrix M need to be estimated. Historical observations Rω, ωεΩ of R may be used to estimate the quantities. However, for a large number of assets, i.e., n>T=|Ω|, using sample averages directly to estimate the mean vector and the covariance matrix do not yield the desired results, because the sample errors tend to be large and also the resulting covariance matrix is rank deficient (positive semidefinite rather than positive definite).
In order to overcome this problem, factor models have been introduced that linearly relate the n-vector of asset returns R to a smaller number k of factors V stated asR=FTV+εwhere F is the k×n matrix of factor loadings and ε, the n-vector of the difference R−FTV, is assumed to be an independently (between its components, respectively, and with respect to V) distributed vector of error terms.
In practice, a factor model is estimated employing pairs of historical observations Rω, Vω and regression analysis to obtain an estimate of F. The estimation results in an asset return process of the formR=FTVω+ε, ωεΩ.Using this asset return process, the covariance matrix can now be obtained asM=FTMvF+D, where MV=[k×k] is the covariance matrix of the factors (or factor returns), D=diag(σi2) is the diagonal matrix of the idiosyncratic risk, and σi2 is the variance of the i-th independent error term εi. The matrix MV is estimated employing historical observations, and because D is diagonal, the resulting covariance matrix M is of full rank (rank(M)=n). The number of parameters to be estimated is much smaller (nk for the factor loadings+k2 for the factor covariances+k for the means) than without imposing the linear factor model ((n(n+1)/2 for the covariances+n for the means), especially when the number of factors k is kept reasonably small.
The mean-variance portfolio optimization problem based on a factor model representation of asset returns,
            max      ⁢                          ⁢              μ        T            ⁢      x        -                  γ        2            ⁢                        x          T                ⁡                  (                                                    F                T                            ⁢                              M                V                            ⁢              F                        +            D                    )                    ⁢      x                  Ax      =      b        ,          l      ≤      x      ≤      h      employing estimates based on historical observations ωεΩ for F, MV, and D is currently the state-of-the art in large-scale portfolio optimization. See R. C. Grinold and R. Kahn, Active Portfolio Management, McGraw-Hill, New York, N.Y., 2000. Commercially available software applications for equity portfolio optimization are typically based on estimated factor models and use mean-variance optimization.
Consequently, known software applications for equity portfolio optimization are limited. It is desirable to also use the factor model for expected utility optimization. The corresponding portfolio optimization problem can be stated asmaxE u (1+(FTVω+ε)Tx)Ax=b,l≦x≦h where the objective function includes the expectation over a discrete random vector (FTVω) and over the continuous random vector ε.
Expected utility maximization problems have been formulated using a sample average approximation based on historical return observations in R. C. Grinold, Mean-variance and scenario-based approaches to portfolio selection, Journal of Portfolio Management, 25(2):10-22, 1999, as
      max    ⁢                  ⁢          1                      Ω                      ⁢                  ∑                  ω          ∈          Ω                                              ⁢              u        ⁡                  (                      1            +                                          R                c                                  ω                  ⁢                                                                          ⁢                  T                                            ⁢              x                                )                                Ax      =      b        ,          l      ≤      x      ≤      h      where the current (subscript c) return observations based on forward looking estimates of mean return and volatility areRciωT=μci+σciziωwhere the z-scores ziω are computed from the return observations Riω by subtracting the historical mean and dividing by the historical standard deviation such that they each have a mean value of zero and a standard deviation of one, and μc and σc are forward looking estimates of asset mean return and volatility.
The sample average model is a good approximation as long as |Ω|>>n, because only then the problem is not rank deficient and statistically viable. However, for large-scale utility maximization problems, where n>|ω|, the sample average approximation based on historical return observations is not a good approximation.
Approximations for a related version of the expected utility maximization problem based on a factor model for asset returns have been described by M. W. Brandt, P. Santa Clara, and R. Valkanov, Parametric portfolio policies: Exploiting characteristics in the cross section of equity returns, Review of Financial Studies, 22(9): 3411-3447, 2004, and by S. De Boer, Factor tilting for expected utility maximization, Journal of Asset Management, 11:31-42, 2010. Brandt et al. (2004) constructed an expected utility maximization model with factor exposures as the decision variables, where the portfolio weights are subsequently derived from the estimated factor loadings and the optimal factor exposures. This model assumes constant factor exposures over time. DeBoer (2010) first calculates the expected utility optimal portfolio weights for a given factor exposure parametrically as a function of possible factor exposures, and then solves the expected utility optimization problem in the factor space. Disadvantageously, both techniques are approximations and appear unable to handle general constraints.
Conceptually, a direct approach might be to employ sampling from the factor model representation of asset returns. Accordingly, let RvεS be a sample of returns of size |S|, sampled independently from the factor model of asset returns R=FTVω+ε. The corresponding sample average approximation of the expected utility maximization problem then is
      max    ⁢                  ⁢          1                      S                      ⁢                  ∑                  v          ∈          S                                              ⁢              u        ⁡                  (                      1            +                                          R                                  v                  ⁢                                                                          ⁢                  T                                            ⁢              x                                )                                Ax      =      b        ,          l      ≤      x      ≤      h      In order to represent the distribution of asset returns accurately and to obtain a problem of full rank, the sample size |S| needs to be very large, i.e., |S|>>n. However, this may be computationally prohibitive for a large number of assets.
It would be desirable to overcome the shortcomings of previous techniques that address the expected utility maximization problem. It is to this end that the present invention is directed. The system and method in accordance with the various embodiments of the present invention proceed differently to solve the expected utility maximization problem by exploiting the structure of the problem. The various embodiments of the present invention thus provide many advantages over conventional large-scale financial portfolio optimization techniques. Accordingly, the various embodiments of the present invention provide a system and method that maximize expected utility optimization in connection with management of a large-scale financial asset portfolio. The foregoing and other objects, features, and advantages of the present invention will become more readily apparent from the following detailed description of various embodiments, which precedes with reference to the accompanying drawing.